Max Neunhöffer

A data structure for a uniform approach to computations with finite groups

We describe a recursive data structure for the uniform handling of permutation groups and matrix groups. This data structure allows the switching between permutation and matrix representations of segments of the input group, and has wide-ranging applications. It provides a framework to process theoretical algorithms which were considered too complicated for implementation such as the asymptotically fastest algorithms for the basic handling of large-base permutation groups and for Sylow subgroup computations in arbitrary permutation groups. It also facilitates the basic handling of matrix groups. The data structure is general enough for the easy incorporation of any matrix group or permutation group algorithm code; in particular, the library functions of the GAP computer algebra system dealing with permutation groups and matrix groups work with a minimal modification.

Computing automorphisms of semigroups

In this paper an algorithm is presented that can be used to calculate the automorphism group of a finite transformation semigroup. The general algorithm employs a special method to compute the automorphism group of a finite simple semigroup. As applications of the algorithm all the automorphism groups of semigroups of order at most 7 and of the multiplicative semigroups of some group rings are found. We also consider which groups occur as the automorphism groups of semigroups of several distinguished types.

COMPUTING MINIMAL POLYNOMIALS OF MATRICES

We present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of an

Condensation of homomorphism spaces

n\times n

Parallelising the computational algebra system GAP

matrix over a finite field that requires

Generalised Sifting in Black-Box Groups

O(n^3)

Sporadic neighbour-transitive codes in Johnson graphs

field operations and O(n) random vectors, and is well suited for successful practical implementation. The algorithm, and its complexity analysis, use standard algorithms for polynomial and matrix operations. We compare features of the algorithm with several other algorithms in the literature. In addition we present a deterministic verification procedure which is similarly efficient in most cases but has a worst-case complexity of

Formulas for primitive idempotents in Frobenius algebras and an application to decomposition maps

O(n^4)

Towards high-performance computational algebra with GAP

. Finally, we report the results of practical experiments with an implementation of our algorithms in comparison with the current algorithms in the {\sf GAP} library.

Space exploration using parallel orbits : a study in parallel symbolic computing

We present an ecient algorithm for the condensation of homomorphism spaces. This provides an improvement over the known tensor condensation method which is essentially due to a better choice of bases. We explain the theory behind this approach and describe the implementation in detail. Finally, we give timings to compare with previous methods.